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Exact results for duality-covariant integrated correlators in $\mathcal{N}=4$ SYM with general classical gauge groups
by Daniele Dorigoni, Michael B. Green, Congkao Wen
This Submission thread is now published as
Submission summary
| Authors (as registered SciPost users): | Congkao Wen |
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| Preprint Link: | scipost_202203_00025v2 (pdf) |
| Date accepted: | 2022-08-30 |
| Date submitted: | 2022-08-07 16:53 |
| Submitted by: | Wen, Congkao |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
We present exact expressions for certain integrated correlators of four superconformal primary operators in the stress tensor multiplet of $\cN=4$ supersymmetric Yang--Mills (SYM) theory with classical gauge group, $G_N$ $= SO(2N)$, $SO(2N+1)$, $USp(2N)$. These integrated correlators are expressed as two-dimensional lattice sums by considering derivatives of the localised partition functions, generalising the expression obtained for $SU(N)$ \mbg{gauge group} in our previous works. These expressions are manifestly covariant under Goddard-Nuyts-Olive duality. The integrated correlators can also be formally written as infinite sums of non-holomorphic Eisenstein series with integer indices and rational coefficients. Furthermore, the action of the hyperbolic Laplace operator with respect to the complex coupling $\tau=\theta/(2\pi) + 4\pi i /g^2_{_{YM}}$ on any integrated correlator for gauge group $G_N$ relates it to a linear combination of correlators with gauge groups $G_{N+1}$, $G_N$ and $G_{N-1}$. These ``Laplace-difference equations'' determine the expressions of integrated correlators for all classical gauge groups for any value of $N$ in terms of the correlator for the gauge group $SU(2)$. The perturbation expansions of these integrated correlators for any finite value of $N$ agree with properties obtained from perturbative Yang--Mills quantum field theory, together with various multi-instanton calculations which are also shown to agree with those determined by supersymmetric localisation. The coefficients of terms in the large-$N$ expansion are sums of non-holomorphic Eisenstein series with half-integer indices, which extend recent results and make contact with low order terms in the low energy expansion of type IIB superstring theory in an $AdS_5\times S^5/\mathbb{Z}_2$ background.
Author comments upon resubmission
List of changes
{\bf Referee 1} The following describes our responses to the referee's three queries.
1) We have added an appendix, as suggested by the referee, in which we review the Laplace difference equation for $SU(N)$ groups. This appendix is referred to before eq. 3.1. Similarly we discuss the Laplace difference equation for $SO$ and $USp$ groups in the same appendix. In these cases the Laplace difference equations are based on the expressions for the functions $B^1(t)$ and $B^2(t)$ given in section 4.2.
2) We have clarified our comments concerning the perturbative expansion given in (2.11). Whereas in the $SU(N)$ case the 't Hooft expansion parameters are $g_{_{YM}}^2 N$ and $N^2$ at both finite $N$ and large $N$, in the case of the other classical groups the appropriate expansion parameters at finite $N$ are different from the parameters at large $N$. The finite $N$ expansion with small $g_{_{YM}}^2 N$ is very far from the standard AdS/CFT correspondence. The analogue of the '''t Hooft'' parameters in this case are the quantitities $a_{G_N}$ defined in (2.4), while the parameter $N^2$ of $SU(N)$ is replaced by $N_{G_N}$ defined in (2.10). These are different choices from those required at large $N$ and large $\lambda$, which are given in (5.4) and (5.5). These are in accord with AdS/CFT duality, which (as the referee points out) is reviewed in appendix C. They are the unique choices that lead to the remarkable properties of the perturbative expansion (2.5). Although we stressed these properties in the bullett points following (2.9) in response to the referee?s comments we have enhanced the wording in these points since the pattern displayed by the perturbative expansion is one of the main points of the paper. We have also stressed that this expansion is different from the 't Hooft expansion at large N considered in section 4, which can be decomposed into terms corresponding to orientable and non-orientable world-sheets.
3) We have also added a reference to arXiv:1908.11276 in the concluding section, as suggested by the referee.
{\bf Referee 2} The following again describes our responses to the referee's three queries.
1) We think our reference to [9] is appropriate and sufficient, given the content of our paper.. That reference concerned the spectral representation of $SU(N)$ integrated correlators, and is equivalent to our lattice representation presented in [1] and [2] i.
For our purposes the lattice representation is somewhat more useful.
This lattice representation was discovered by exploring the perturbative and instanton sectors of the localised correlators defined in [3], which suggested the expression is given by a formal sum of Eisenstein series. In the present paper the key to obtaining the lattice representation for any classical gauge group was again a careful analysis of perturbative contributions (based on [10]) as well as instanton contributions, which are more difficult to determine. This led to the more subtle formal sum of Eisenstein series in (1.8), which satisfy the GNO S-duality constraints and led to the lattice representation (1.3). The expression in terms of Eisenstein series is mentioned in the abstract since it provides a novel mathematical representation of GNO duality that plays a crucial r\^ole in our derivation of the lattice representation.
2) As the referee implies, the functions $B^1$ and $B^2$ were indeed motivated by the perturbation expansion of the integrated correlator. They are constructed in a manner that preserves GNO duality. As described in section 4.3, it is a non-trivial task to extract the instanton data that is consistent with these sums of Eisenstein series. These functions are indeed Borel transforms of the correlator from which the Laplace difference equations and the convergence properties of the correlators in various limits can be determined. The referee asks for an explanation of what $B^1$ and $B^2$ are physically. These expressions emerge from a mathematical analysis of the integrated correlator and do not have an independent physical interpretation.
3) The small comment in the conclusion about the hope of finding a lattice representation of the $\partial_m^4 \log Z|_{m=0}$ correlator is based partly on the large-N expansion explored in Chester et al 2008.02713. There it was shown that the coefficients of integer powers of $1/N$ are sums of 'generalised Eisenstein series' that satisfy inhomogeneous Laplace eigenvalue equations. These do indeed have lattice representations as shown in 0807.0389 and earlier papers. It is not clear to us why the referee suggests that the results in Chester et al suggest the contrary.
Published as SciPost Phys. 13, 092 (2022)